Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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When does a polynomial in $GF$ have a multiplicative inverse?

When does a polynomial in $GF$ have a multiplicative inverse? Are there values of $n$ such that all polynomials in $GF(n)$ have multiplicative inverses? EDIT: To address the comments, I mean: All ...
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1answer
99 views

Are all functions on vectors in $GF(2^n)$ representable as polynomial functions?

In $GF(2)$, any function from an n-dimensional vector to a number is equal to a polynomial function of n variables. (See proof below.) Question: Is this true for other $GF$, especially $GF(2^8)$? ...
2
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1answer
109 views

What does it mean for an algebraic integer to have an abelian galois group?

What does it mean for an algebraic integer to have an abelian galois group ? I thought all algebraic integers had an abelian galois group ? Beware, I'm new to galois theory. Maybe examples and ...
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2answers
134 views

Question about algebraic closures

Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. Let $K = \mathbb{Q}(\sqrt{d})$ and $\overline{K}$ defined to be the algebraic closure of $K$. Is it true that $\overline{K} \cong ...
2
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2answers
122 views

Embedding ordered number fields into $\Bbb R$

Consider a number field $K$ equipped with a total order $\le$. Is there always a field homomorphism $\phi:K \rightarrow \Bbb R$ which respects the total order, i. e. with $\phi(x)>\phi(y)$ for ...
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2answers
126 views

Normal field extension separable over its fixed field

Let us have a field $K\supseteq E$ and $G$ be its group of automorphisms over $E$. Let the fixed field of $G$ be $K^G$. I would like to show that $K$ is separable over $K^G$. I know that for ...
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388 views

Proof that a finite separable extension has only finite many intermediate fields

Let $E/F$ be finite separable extension. Is there any proof of the fact that there are only finitely many intermediate fields without using primitive element theorem or fundamental theorem of Galois ...
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2answers
237 views

Two different definitions of separable polynomial

This is from A field guide to Algebra by Antoine Chambert Loir. A polynomial $P \in K[X]$ is separable if and only if its roots are in an algebraic closure of $K$ are simple. Here is ...
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131 views

Application of Hilbert 90 for Finite Fields

Let $k = \mathbb{F}_{p^n} = \mathbb{F}_q$ finite field of $q = p^n$ and $[K:k]=2$ Galois extension of degree 2. Then $K = \mathbb{F}_{q^2} = \mathbb{F}_{(p^n)^2} = \mathbb{F}_{p^{2n}}$. It is ...
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2answers
298 views

Field of characteristic 0 such that every finite extension is cyclic

I am trying to construct a field $F$ of characteristic 0 such that every finite extension of $F$ is cyclic. I think that I have an idea as to what $F$ should be, but I am not sure how to complete the ...
3
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1answer
82 views

Finding degree of the extension [duplicate]

Is it true that the degree of extension $\mathbb Q(\sqrt {2},\sqrt {3},\sqrt {5},\dotsc,\sqrt {p_n}) / \mathbb Q$ is $2^n$ where $p_n$ is the $n$th prime number. If so, how to prove this? My idea is ...
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247 views

Necessarily complex analytic proofs in algebra.

Does anyone know of an example where complex analysis is necessary to prove something in algebra? I would be particularly interested in results from group theory or Galois theory. In an ideal ...
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1answer
61 views

Norm -1 in the extension $\,E[i]/E\,$ , where $\,E=\Bbb Q(\zeta)\,,\,\, \zeta^5 = 1$

Denote by $\zeta = \exp(2\pi i/5)$ the primitive root of unit of order 5 ($\zeta^5=1, \zeta \ne 1$). Let $E = \mathbb{Q}[\zeta]$. Then $i = \sqrt{-1} \notin E$. Let $L = E[i]$. We want to show that ...
4
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5answers
241 views

Field extension of composite degree has a non-trivial sub-extension

Let $E/F$ be an extension of fields with $[E:F]$ composite (not prime). Must there be a field $L$ contained between $E$ and $F$ which is not equal to either $E$ or $F$? To prove this is true, it ...
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3answers
633 views

Finding the degree of a field extension over the rationals

Let $\alpha = \sqrt{\sqrt{2}+\root 4 \of 2}$, $\beta =\sqrt{\sqrt{2}- \root 4 \of 2}$, $\gamma = \sqrt{-\sqrt{2}+i\root 4 \of 2}$ and $\delta = \overline{\gamma}=\sqrt{-\sqrt{2}-i\root 4 \of 2}$. Let ...
8
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2answers
640 views

closed-form expression for roots of a polynomial

It is often said colloquially that the roots of a general polynomial of degree $5$ or higher have "no closed-form formula," but the Abel-Ruffini theorem only proves nonexistence of algebraic ...
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1answer
113 views

Is $\mathbb Q\left(\sqrt{5+\sqrt{5}}, \sqrt{5-\sqrt{5}}\right)$ a simple extension?

I know that it should be because is finite and separable ($\rm{char}(\mathbb Q)=0$). However I'm having some trouble in finding the primitive element. First of all, given the irreducible polynomial ...
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400 views

Intersection of Cyclotomic Fields

How would I prove that $\mathbb{Q_m} \cap \mathbb{Q_n} = \mathbb{Q_{(m, n)}}$ (here $\mathbb{Q_n}$ denotes the $n$th cyclotomic field)? I already know of a solution involving the fact that given two ...
6
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5answers
335 views

why are subextensions of Galois extensions also Galois?

An algebraic extension of fields $L|K$ is defined to be a Galois extension iff the set of elements of $L$ invariant under the action of $Aut_K L$ is $K$. Apparently in the sequence of field ...
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1answer
520 views

Inverse Galois problem for small groups

I am looking for a list of all small groups (maybe order $\leq 20$) realized as the Galois groups of a polynomial over $\mathbb{Q}$, with proof. Any idea where I could find these? Partial answers or ...
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0answers
51 views

Exist automorphism from an element to its conjugate

I was asked to prove the galois group of a given normal extension is non-abelian. My original solution was to use isomorphism extension theorem but that was not taught in class. So, in my new attempt ...
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1answer
898 views

Existence of irreducible polynomial of arbitrary degree over finite field without use of primitive element theorem?

Suppose $F_{p^k}$ is a finite field. If $F_{p^{nk}}$ is some extension field, then the primitive element theorem tells us that $F_{p^{nk}}=F_{p^k}(\alpha)$ for some $\alpha$, whose minimal polynomial ...
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2answers
400 views

Intermediate fields of splitting field

I'm trying to find the intermediate fields of the extension $\mathbb Q\big /\mathbb Q(\alpha)$, where $\alpha = \sqrt{7+\sqrt{13}}$. To do so I've tried to use the Galois correspondence. I've already ...
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1answer
478 views

Finding degree of a splitting field

I am trying to find the degree of the splitting field of the irreducible polynomial $f(x)=x^4-14x^2+36\in\mathbb Q[x]$. The roots of $f$ are $\left\{\pm\sqrt{7\pm\sqrt{13}}\right\}$, so the splitting ...
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1answer
199 views

Galois group of a degree 8 polynomial

Can $D_{8}$ be the Galois group of $f$ where $f$ is a degree 8 polynomial? I was thinking that the Galois group must act transitively on the roots of $f$ (if $f$ is irreducible). In that case ...
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2answers
306 views

Finding the Galois group of $\mathbb Q (\sqrt 5 +\sqrt 7) \big/ \mathbb Q$

I know that this extension has degree $4$. Thus, the Galois group is embedded in $S_4$. I know that the groups of order $4$ are $\mathbb Z_4$ and $V_4$, but both can be embedded in $S_4$. So, since I ...
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2answers
84 views

Isomorphism with cyclotomic extension

Let $K$ be a field with $\mbox{char}(K)=0$. I know that if $\xi$ is a primitive $n$-th root of the unity and $K(\xi)\big/K$ is a cyclotomic extension, then $\mbox{Gal}\left(K(\xi)\big/K\right)$ is ...
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3answers
905 views

Quintic polynomial with Galois Group $A_5$

A recent question asks what makes degree 5 special when considering the roots of polynomials with integer coefficients etc. One answer is that the Galois Group of $S_5$ is not solvable. What I am ...
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1answer
76 views

Galois correspondence between normal groups and normal extensions

I do know that, given an extension $F\big/K$, if there is a normal intermediate extension, the corresponding subgroup of $\mbox{Gal}\left(F\big/K\right)$ is normal. The problem is that I don't see why ...
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Galois group acts transitively

The question I am dealing with is: Let F be a field, $f(x)\in F[x]$ be irreducible and let $N/F$ be normal field extension. Let $$f(x) = g_1(x) \cdot \dots \cdot g_r (x)$$ be the factorization of ...
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1answer
277 views

Doesn't Abhyankar's lemma contradict faithful flat descent (no, but I'm confused)

Let $O_K$ be a dvr with fraction field $K$. Let $L/K$ be a tamely ramified finite Galois extension. Then, Abhyankar's Lemma implies that there exists a finite Galois extension $K^\prime/K$ such that ...
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1answer
132 views

Is a finite field extension of a imperfect field imperfect

Let $K$ be a imperfect field. Let $L/K$ be a finite field extension. Is $L$ imperfect? Suppose that $L/K$ is separable. Is $L$ imperfect? Suppose that $L/K$ is Galois. Is $L$ imperfect? I'm ...
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degree of the extension galois

I have a problem with the solution of the tasks of abstract algebra.Help please. $F=({\bf Q};+;\cdot),K=({\bf R},+;\cdot)$. Determine the degree of the extension $F_K^* (\sqrt 2,\sqrt 3):F]$. ...
4
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1answer
226 views

Trace and Norm of a separable extension.

If $L | K$ is a separable extension and $\sigma : L \rightarrow \bar K$ varies over the different $K$-embeddings of $L$ into an algebraic closure $\bar K$ of $K$, then how to prove that ...
4
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1answer
374 views

Square roots of integers and cyclotomic fields

For every $ N \in \mathbb Z$ there exists an integer $n$ such that $ \sqrt N \in \mathbb Q(\zeta_n)$. I am struggling where to start this question, please suggest me few hints.
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673 views

Extensions of degree two are Galois Extensions.

This question from Allan Clark's "Elements of Abstract Algebra" Show that an extension of degree 2 is Galois except possibly when the characteristic is 2. What is the case when the characteristic is ...
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1answer
123 views

Galois Group of Polynomial

I would like to compute the Galois group of the Polynomial $f(x)=x^5-5x^4 +10 x^3 - 10 x^2 - 135 x + 131\in\mathbb{Q}[x] $ I already know that it is irreducible in $\mathbb{Q}[x]$ via Eisenstein's ...
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1answer
118 views

Isomorphism of Galois groups induces an isomorphism on decomposition groups?

This is a follow up question to this. Say we have Galois extensions $L ,M$ of $\Bbb{Q}$ with $L \cap M = \Bbb{Q}$ and consider the composite $K = LM$. I want to prove that $$\begin{array}{cccccc} f : ...
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1answer
148 views

Do we have such a direct product decomposition of Galois groups?

Let $L = \Bbb{Q}(\zeta_m)$ where we write $m = p^k n$ with $(p,n) = 1$. Let $p$ be a prime of $\Bbb{Z}$ and $P$ any prime of $\mathcal{O}_L$ lying over $p$. Notation: We write $I = I(P|p)$ to denote ...
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1answer
86 views

Cardano's Formulas help

I am working on solving this cubic: $x^3 +x^2 - 2 = 0$ using Cardano's explicit formulas: $$ A = \sqrt[3]{{-27\over 2}q + {3 \over 2} \sqrt{-3D}} \qquad B = \sqrt[3]{{-27\over 2}q - {3 \over 2} ...
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Adjoining a square root of the discriminant of an irreducible cubic.

Here is an exercise. Let $\delta$ be a square root of the discriminant of $P$, an irreducible cubic polynomial over a field $K$ with characteristic not equal to 2. Show that $P$ is irreducible ...
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Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider ...
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The Galois group of $f(t)=t^3-x_1t^2+x_2t-x_3$

I'm trying to solve this question: Let $F[x_1, x_2,x_3]$ be a polynomial ring in $x_1, x_2,x_3$ over a field $F$. Let $K=F(x_1,x_2,x_3)$ be the field of rational functions (i.e., the ...
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3answers
209 views

Existence of Galois group of order 8 as $\mathbb Z_2\times \mathbb Z_4$

I'm trying to find a Galois group of $\mathbb Z_2\times \mathbb Z_4$ but, first, I'd like to know if it does exist. I know that a Galois group of order $8$ must be a subgroup of the symmetric group of ...
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1answer
103 views

Steps missing in the proof $[L:K]=n!\ \Longrightarrow \ f\text{ is irreducible}$

Consider the following THEOREM Let $f\in K[X]$ have degree $n$ and splitting field $L/K$. Then we have $$[L:K]=n!\ \Longrightarrow \ f\text{ is irreducible}$$ and its Proof $\ $ Suppose ...
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1answer
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Separability by surjectivity of Frobenius Endomorphism

I'm trying to prove the following statement: Let $F$ be a finite field of prime characteristic $p$ and let $E$ be the field generated by $F$ and the elements $\{t^{1/p},n \geq 1\}$, where $t$ ...
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1answer
141 views

Unconventional Galois theory in Qual

This seems to be an uncommon question that one may see in Qual Algebra exams, and I don't think if it was a wise choice to put this kind of questions in Qual. I consider it as a hard Galois theory ...
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214 views

$[L:K]=n!\ \Longrightarrow \ f$ is irreducible and $\text{Gal}(L/K)\cong S_n.$

How do I go about proving the following theorem ? Let $f\in K[T]$ have degree $n$ and splitting field $L/K$. Then we have $$ [L:K]=n!\ \Longrightarrow \ f\text{ is irreducible and Gal}(L/K)\cong ...
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299 views

Irreducibility in finite field implies irreducibility in $\mathbb{Z}$?

Let $p$ be a prime, $f$ be a polynomial of $\mathbb{Z}[x]$. Suppose that $f$ is irreducible in $\mathbb{F}_p[x]$. My question is : Is $f$ irreducible in $\mathbb{Z}[x]$ ? This question is ...
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2answers
81 views

Base of a Galois extension

For a field $k$, I know that $k(x_1,\cdots,x_n)/k(s_1,\cdots,s_n)$ is a finite Galois extension with Galois group $S_n$ where $s_i$ is an elementary symmetric polynomial. Thus its dimension is $n!$. ...